Comprehensive Research Toward the Global Theory for the System of Nonlinear Partial Differential Equations

非线性偏微分方程组整体理论的综合研究

基本信息

批准号：
10304012
负责人：
NISHIDA Takaaki
金额：
$ 17.66万
依托单位：
KYOTO UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (A).
财政年份：
1998
资助国家：
日本
起止时间：
1998 至 2000
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10304012/
关键词：
Nonlinear PDE Global bifurcation structure Bifurcation from degenerate singular points Dynamical system Computer assisted proof Navier-Stokes equation Heat convection problem Taylor-Couette problem 縮約系の大域的解析精度保証付き数値計算粘性的衝撃波燃焼合成反応

项目摘要

1. Heat convection problem :In order to investigate the global structure of the solution space of the nonlinear PDE's and to treat the global bifurcation curves in it, we worked on the analytical method combined with the computational analysis and computer assisted proof. We proposed criterions to prove the existence of solutions which correspond to parameter values as computer assited proof. Using the method we showed the existence of global bifurcation curves on which the roll-type solutions exist that correspond to large Rayleigh numbers.In the case of 3-dimension we investigated numerically the pattern formation of roll-type, rectangle-typpe and hexagonaltype solutions and their stability, and we clarified the global bifurcation diagram which is not seen from the local bifurcation theory.2. Taylor problem :We considered the stability of Couette flow when the two cylinder rotate in the opposite directions. It is reduced to the eigenvalue problem for the system of ordinary differenti … More al equations and it can be treated by our computer assisted proof to see the exact critical Taylor number, at which the stationary or Hopf bifurcation occurs. The bifurcation point with multiplicity is one of our future subject.3. The existence theorem for stationary solution of Navier-Stokes equation is proved by our numerical verification method at least for small Reynols number.4. Dynamical systems :We know that when the degeneracy of singular points of vector field increases, the behavior of dynamics becomes more complex and the global phenomena become more included. We investigated the singular point with codimension 3 and proved analytically that the hetero-clinic cycle bifurcates and also chaotic attractor does.5. For the 3-dimensional exterior problem of stationary Navier-Stokes equation, we introduced a real interpolation of Morrey spaces to solve N-S equation and succeeded to construct the exterior stationary solution and to prove its stability without the unnatural zero net force conditions. Less

1。热连接问题：为了研究非线性PDE的解决方案空间的全局结构并处理其中的全局分叉曲线，我们使用了分析方法与计算分析和计算机辅助证明相结合。我们提出的标准是证明存在与参数值作为计算机分配的证明的解决方案的存在。使用该方法，我们显示了存在与大雷利数字相对应的卷型解决方案存在的全球分叉曲线的存在。在三维的情况下，我们在数值上研究了滚动型，矩形式和六边形型解决方案的模式形成，并从全球范围内阐明了bifurcation nifife nifif。泰勒问题：当两个气缸朝相反的方向旋转时，我们考虑了Couette流的稳定性。它减少为普通差异系统的特征值问题……更多的方程式，我们的计算机辅助证明可以对其进行处理，以查看确切的关键泰勒数字，在这种情况下发生固定或霍普夫分叉发生。具有多重性的分叉点是我们未来的主题之一。3。 Navier-Stokes方程的固定解的存在理论通过我们的数值验证方法证明至少对于小雷诺数字。4。动态系统：我们知道，当对向量场的奇异点的退化护理增加时，动力学的行为变得更加复杂，全局现象变得更加包括在内。我们研究了具有编码3的奇异点，并在分析上证明了异质 - 临床周期分叉以及混乱的吸引子。5。对于固定的Navier-Stokes方程的三维外部问题，我们引入了Morrey空间的真实插值来求解N-S方程，并成功地构建了外部固定溶液，并证明其稳定性而没有非自然的零净力条件。较少的